Escherization

Over his life, the Dutch graphic artist M.C. Escher created over
a hundred ingenious tesselations in the plane.
Some were simple and geometric, used as prototypes for more complex
endeavors. But in most the tiles were recognizable animal forms
such as birds, fish and reptiles.
The definitive reference on Escher's divisions of the plane is Doris
Schattschneider's Visions of Symmetry, now in a second edition.
A good place to view some of Escher's work online is
the World of Escher.
Be sure also to check out these examples from Escher's notebook:
E25,
E70, and
E72.

Escher was able to discover such tilings through a combination of
natural ability and sheer determination. Can we automate the discovery
of tilings by recognizable motifs? More formally, we pose the
Escherization problem:

Given a shape S, find a new shape T such that:

T is as close as possible to S; and

Copies of T fit together to form a tiling of the plane.

We have developed an algorithm that can produce reasonable solutions to
the Escherization problem. It is based on three large components:

A parameterized space of tilings. We develop a parameterization
of the space of isohedral tilings, a family of tilings that
can express the designs Escher created. Every possible tile
shape boils down to a sequence of real numbers. An interactive
tool lets us modify these numbers in a natural way (corresponding
to Escher manipulating drawings by hand). More importantly, we
can direct the computer to search through possible sequences of
numbers, trying to find good tilings by brute force.

A comparison function for shapes. We can dial up a collection
of tile shapes using the parameterization above. We need to know
how well each of these tiles approximates the goal shape provided
by the user. We use a metric from the computer vision
literature that efficiently finds the L2 distance
between two polygons. Our objective then becomes to locate the
particular tile shape that compares most favourably to the goal
shape.

An optimization algorithm. We need a meaningful strategy
for sorting through the large shape of possible tile shapes for
the best one (the one that's closest to the goal shape). We use
a simulated annealing algorithm from Numerical Recipes.
When suitably tuned, it tends to find good tile shapes without
getting stuck in objectionable local minima.

Here are some images produced using Escherization. Click on each
one for a full-size version.

A Plague of Frogs

Dogs; Dogs Everywhere

Tea-ssellation

Tux-ture Mapping

Twisted Sisters

Wiener Dog Art

Dihedral Escherization

Escher also created a number of dihedral tilings: designs featuring
two different shapes. He was fond of these designs, since with more
than one shape it was possible to tell a story, to have the shapes
complement each other somehow.

The Escherization problem can be adapted in an obvious way to two
goal shapes. We can also extend the space of tilings by including
in the parameterization a path that splits an isohedral tile shape into
two pieces. We compare the two pieces to the goal shapes and run
the optimization as before, attempting to minimize the maximum of
the two comparisons.

Gödel, Bach (Braided): An Eternal Escherization

Funky Chickens

The Owl and the Pussycat

Pen/Rose Tiling

Rembrandt and Mrs. van Rijn

Strange 'Tractors

Dihedral Escherization can also be exploited to create designs in
the style of Escher's
Sky and
Water. We need to restrict
the search to a narrower class of possible tiling types (what Dress
calls “Heaven and Hell Patterns”). Once the desired
tiling has been discovered the rest of the process is fairly easy,
since we already have an association between a realistic, user-supplied
goal shape and an abstract, more geometric tile shape. We just need
to morph between those two extremes. Here's one example, based on
Rembrandt and Mrs. van Rijn above.

Aperiodic Escherization

The Penrose tilings P2 (kites and darts) and P3 (thin and thick rhombs)
can also be parameterized and fed to the Escherization system. It's
much harder to discover satisfying results, because of the idiosyncratic
shapes of the Penrose tiles. It's exciting to be able to produce these
aperiodic pictures – Penrose and Escher were friends, but sadly
Escher passed away before Penrose discovered P2 and P3. Penrose has
asked what sorts of designs Escher would have been able to create from
these tilings.

A Walk in the Park

Busby Berkeley Chickens

The Pentalateral Commission

You might enjoy playing with the edges of Penrose tiles to see whether
you can produce recognizable shapes. I've created a
Java applet
that lets you do so interactively.

Non-Euclidean Escherization

Escher made four wonderful Circle Limit patterns, designs
based on non-Euclidean geometry. Escher didn't have the mathematical
background to manipulate hyperbolic patterns symbolically, but he certainly
had the intuition necessary to create pictures of them.

Our Escherization algorithm cannot easily be translated into non-Euclidean
geometry, for deep reasons having to do with the notion of shape, as used
in the shape comparison metric. However, for some isohedral tiling types
it is possible to transfer the output of Escherization from
the Euclidean plane to the hyperbolic plane or to the surface of a sphere.
The process is a kind of non-linear warp of a piece of the Escherized tiling.

This file contains specifications for the representation and
drawing of the 81 isohedral tilings realizable by unmarked tiles.
Among other things, it provides the incidence symbol, a simple
colouring, and a set of rules for deriving aspect transform
matrices.

These two files together provide a script for navigating the
tiling vertex parameterizations of the isohedral tiling types.
params.py contains a function for each unique parameterization,
and can be used as a source from which to build your own library
for parameterizing legal isohedral tiling polygons.

I have also made a Java applet that allows you to explore the space of
Penrose tile shapes interactively. It's a fun way to play with this
rather mysterious tiling. Most of the effort related to aperiodic tilings
seems to be related to their combinatorial properties, without considering
the actual shapes that can be produced. You can try the applet
here.

All final tilings are copyright 2000 by Craig S. Kaplan. You are
free to use them for personal and non-commercial purposes.
Please check with me about any other uses.